AI中文摘要
假设 $\mu$ 和 $\nu$ 是 $\mathbb{R}$ 上的概率测度，满足 $\mu \leq_{cx} \nu$。设 $a$ 和 $b$ 是 $\mathbb{R}$ 上的凸函数，且 $a \geq b \geq 0$。我们感兴趣的是寻找 $$\sup_{\mathbf{M}} \sup_{\tau} \mathbb{E}^{\mathbf{M}} \left[ a(X) I_{ \{ \tau = 1 \} } + b(Y) I_{ \{ \tau = 2 \} } \right] $$ 其中第一个上确界取遍所有一致模型 $\mathbf{M}$（即过滤概率空间 $(\Omega, \mathbf{F}, \mathbb{F}, \mathbb{P})$，使得 $Z=(z,Z_1,Z_2)=(\int_{\mathbb{R}} x \mu(dx) = \int_{\mathbb{R}} y \nu(dy), X, Y)$ 是一个 $(\mathbb{F},\mathbb{P})$ 鞅，且在 $\mathbb{P}$ 下 $X$ 服从分布 $\mu$，$Y$ 服从分布 $\nu$），第二个上确界中的 $\tau$ 是取值于 $\{1,2\}$ 的 $(\mathbb{F},\mathbb{P})$ 停时。我们的贡献首先是刻画并简化对偶问题，其次是在对测度 $\mu$ 和 $\nu$ 的一些结构假设（即 $\mu$ 和 $\nu$ 是绝对连续的概率测度且满足分散性假设）下完全求解该问题。一个关键发现是，由 $Z$ 生成的过滤的标准设定不足以定义最优模型，需要额外的随机化。即使边际分布 $\mu$ 和 $\nu$ 是无原子的，这一结论仍然成立。该问题可解释为：在给定同时到期的欧式期权价格的情况下，寻找具有两个可能行权日的百慕大期权价格的稳健或无模型无套利上界。
英文摘要
Suppose $μ$ and $ν$ are probability measures on $\mathbb{R}$ satisfying $μ\leq_{cx} ν$. Let $a$ and $b$ be convex functions on $\mathbb{R}$ with $a \geq b \geq 0$. We are interested in finding $$\sup_{\mathbf{M}} \sup_τ \mathbb{E}^{\mathbf{M}} \left[ a(X) I_{ \{ τ= 1 \} } + b(Y) I_{ \{ τ= 2 \} } \right] $$ where the first supremum is taken over consistent models $\mathbf{M}$ (i.e., filtered probability spaces $(Ω, \mathbf{F}, \mathbb{F}, \mathbb{P})$ such that $Z=(z,Z_1,Z_2)=(\int_{\mathbb{R}} x μ(dx) = \int_{\mathbb{R}} y ν(dy), X, Y)$ is a $(\mathbb{F},\mathbb{P})$ martingale, where $X$ has law $μ$ and $Y$ has law $ν$ under $\mathbb{P}$) and $τ$ in the second supremum is a $(\mathbb{F},\mathbb{P})$-stopping time taking values in $\{1,2\}$. Our contributions are first to characterise and simplify the dual problem, and second to completely solve the problem under some structural assumptions on the measures $μ$ and $ν$ (namely that $μ$ and $ν$ are absolutely continuous probability measures that satisfy the Dispersion Assumption). A key finding is that the canonical set-up in which the filtration is that generated by $Z$ is not rich enough to define an optimal model and additional randomisation is required. This holds even though the marginal laws $μ$ and $ν$ are atom-free. The problem has an interpretation of finding the robust, or model-free, no-arbitrage bound on the price of a Bermudan option with two possible exercise dates, given the prices of co-maturing European options.
